Interaction and Evolution in Classical and Quantum Physics, and Indefinite Causal Structure

Abstract

In classical mechanics, the representations of dynamical evolutions of a system and those of interactions the system can have with its environment are different vector fields on the space of states: evolutions and interactions are conceptually, physically and mathematically different in classical physics, and those differences arise from the generic structure of the very dynamics of classical systems ("Newton's Second Law"). Correlatively, there is a clean separation of the system's degrees of freedom from those of its environment, in a sense one can make precise. I present a theorem showing that these features allow one to reconstruct the entire flat affine 4-dimensional geometry of Newtonian spacetime---the dynamics is inextricably tied to the underlying spacetime structure. In quantum theory (QT), contrarily, the representations of possible evolutions and interactions with the environment are exactly the same vector fields on the space of states ("add another self-adjoint operator to the Hamiltonian and exponentiate"): there is no difference between "evolution" and "interaction" in QT, at least none imposed by the structure of the dynamics itself. Correlatively, in a sense one can make precise, there is no clean separation of the system's degrees of freedom from those of the environment. Finally, there is no intrinsic connection between the dynamics and the underlying spacetime structure: one has to reach in and attach the dynamics to the spacetime geometry by hand, a la Wigner (e.g.). How we distinguish interaction from evolution in QT and how we attach the dynamics to a fixed underlying spacetime structure come from imposing classical concepts foreign to the theory. Trying to hold on to such a distinction is based on classical preconceptions, which we must jettison if we are to finally come to a satisfying understanding of QT. These observatons offer a way to motivate and make sense of, inter alia, the idea of indefinite causal structures.